1 (a)
Prove that f(z)=x

^{2}-y^{+2ixy is analytic and find f'(z).}
5 M

1 (b)
Find the Fourier series expansion for f(x)=|x|, in (-π, π).

5 M

1 (c)
Using laplace transform solve the following differential equation with given condition \[ \dfrac {d^2y}{dt^2} +y=t, \] given taht y(0)=1&y'(0)=0.

5 M

1 (d)
If \[ \overline A = \nabla (xy+ yz+ zx), \ find \ \nabla\cdot \overline A \ and \ \nabla \times \overline A \]

5 M

2 (a)
\[ if \ L[J_0 (t) ] = \dfrac {1}{\sqrt{s^2+1}} \] prove \[ \int^\infty_0 e^{-6t} t J_0 (4t) dt = 3/500 \]

6 M

2 (b)
Find the directional derivative of &straighthi;=x

^{4}+y^{4}+z^{4}at (1,-2,1) in the direction of AB where B is (2,6,-1). Also find the maximum directional of ϕ at (1,-2,1).
6 M

2 (c)
Find the Fourier series expansion for f(x)=4-x

^{2}, in (0,2) Hence deduce that \[ \dfrac {\pi^2}{6}=\dfrac {1}{1^2} + \dfrac {1}{2^2}+ \dfrac {1}{3^2} \cdots \ \cdots \]
8 M

3 (a)
Prove that \[ J_{1/2}(x) = \sqrt{\dfrac {2}{\pi x}} \sin x \]

6 M

3 (b)
Using Green's theorm evaluate \[ int_c (2x^2-y^2)dx + (x^2+y^2) dy where 'c' is the boundary of the surface enclosed by the line x=0, y=0, x=2, y=2.

6 M

3 (c)
i) Find Laplace Transform of \[ e^{-\pi} \int^t_c u \sin 3u \ du

ii) Find Laplace Transform of \[ \dfrac {d}{dt} \left ( \dfrac {1-\cos 2t}{t} \right )

ii) Find Laplace Transform of \[ \dfrac {d}{dt} \left ( \dfrac {1-\cos 2t}{t} \right )

8 M

4 (a)
Obtain complex form of Fourier series for the function f(x)=sin ax in (-?, ?), where a is not an integer.

6 M

4 (b)
Find the analytic function whose imaginary part is \[ v=\dfrac {x}{x^2+y^2} + \cos h \ y\cdot \cos x\]

6 M

4 (c)
Find inverse Laplace Transform of following \[ i) \ \log \left [\dfrac {s^2+a^2}{\sqrt{s+b}} \right ] \\
ii) \ \dfrac {1}{s^3 (s-1)} \]

8 M

5 (a)
Obtain half-range cosine series for f(x)=x(2-x) in 0

6 M

5 (b)
Prove that \[ \overline{F} = \dfrac {\overline r}{r^3} \] is both irrotational and solenoidal.

6 M

5 (c)
Show that the function u=sin x cosh y+2 cos x sinh y+ x

^{2}-y^{2}+4xy satisfies Laplace's equation and find it corresponding analytic function.
8 M

6 (a)
Evaluate by Stoke's theorem \[ \int_c (xy \ dx + x y^2 \ dy) \] where C is the square in the xy-plane with vertices (1,0), (0,1), (-1,0) and (0,-1).

6 M

6 (b)
Find the bilinear transformation, which maps the points z=1,1,? onto the points w=-i, -1, i.

6 M

6 (c)
Show that the general solution of \[ \dfrac {d^2 y }{d x^2} + 4x^2 y=0 \ is \ y=\sqrt{z} [A J_{1/4} (x^2) + B \ J_{-1/4} (x^2)] \] where A and B are constants.

8 M

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